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Unfortunately, the padic_regulator method does not handle the descent_second_limit argument, which can not be directly increased :

sage: reg = x.padic_regulator(p,prec=7, descent_second_limit=20); reg
TypeError: padic_regulator() got an unexpected keyword argument 'descent_second_limit'

A workaround can be the following,

There is also a problem when computing the gens of x as well:

sage: x.gens()
RuntimeError: Unable to compute the rank, hence generators, with certainty (lower bound=0, generators found=[]).  This could be because Sha(E/Q)[2] is nontrivial.
Try increasing descent_second_limit then trying this command again.

But you can compute the gens of x with a higher descent_second_limit argument:

sage: x.gens(descent_second_limit=20)
[(94248114183274/1566972225 : 912499854364645191562/62028595526625 : 1)]

Fortunately the gens are cached so further calls of this method will not require a higher descent_second_limit argument to be set:

sage: x.gens()
[(94248114183274/1566972225 : 912499854364645191562/62028595526625 : 1)]

In particular, now you can compute the padic_regulator:

sage: reg = x.padic_regulator(p,prec=7); reg
2*7 + 5*7^2 + 6*7^3 + 5*7^4 + 3*7^5 + 3*7^6 + O(7^7)

I agree that this is tricky, and i consider your problem as a bug.